Question

Consider the following probability distributions.
(A) Normal distribution
(B) Binomial distribution
(C) Poisson distribution
(D) F-distribution
(E) Chi-square distribution
In which of the above distributions mean and variance are equal:

• (A) only.
• (B) only.
• (C) only.
• (D) and (E) only.

Hint:

Recall that for Poisson(lambda), both mean and variance equal lambda.

Answer 1

The Correct Option is C

Step 1: For the Poisson distribution with parameter \( \lambda \), the mean is \( \lambda \) and the variance is also \( \lambda \), so they are always equal.
Step 2: For the Binomial distribution, mean is \( np \) and variance is \( np(1-p) \), which are equal only if \( p = 0 \), not in general.
Step 3: The Normal distribution has two independent parameters, mean \( \mu \) and variance \( \sigma^2 \), which need not be equal. The F and Chi-square distributions also have mean and variance formulas that differ from each other in general.
Step 4: So only the Poisson distribution, (C), always has mean equal to variance, option 3.